A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 36, 8, 1, 1, 1, 120, 576, 216, 16, 1, 1, 1, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1

Offset: 0

Author

Alois P. Heinz, Jul 29 2013

Keywords

• nonn
• tabl

Comments

A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k. A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.

A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k. A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.

A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n. A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.

A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k. A(2,3) = 36:

(1,2,2)-(1,1,2) (0,1,1)-(0,0,1)

/ X \ / X \

(2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).

\ X / \ X /

(2,2,1) (2,1,1) (1,1,0) (1,0,0)

A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1. A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

Examples

			Square array A(n,k) begins:
  1, 1,  1,    1,       1,           1, ...
  1, 1,  2,    6,      24,         120, ...
  1, 1,  4,   36,     576,       14400, ...
  1, 1,  8,  216,   13824,     1728000, ...
  1, 1, 16, 1296,  331776,   207360000, ...
  1, 1, 32, 7776, 7962624, 24883200000, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..36, flattened

Crossrefs

Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.

Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.

Main diagonal gives: A036740.

Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

Programs

Maple

A:= (n, k)-> k!^n:
seq(seq(A(n,d-n), n=0..d), d=0..12);

Formula

A(n,k) = (k!)^n.

A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.

A(n,k) = k! * A(n-1,k) for n>0, A(0,k) = 1.

G.f. of column k: 1/(1-k!*x).